Dirichlet problem for a linear elliptic equation in unbounded domains with <span class="mathjax-formula">$L^2$</span>-boundary data

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

J. C ^HABROWSKI

Dirichlet problem for a linear elliptic equation in unbounded domains with L

²

-boundary data

Rendiconti del Seminario Matematico della Università di Padova, tome 71 (1984), p. 287-328

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Elliptic Equation

in Unbounded Domains ^with L2 - Boundary ^Data.

J. CHABROWSKI

1. Introduction.

The main purposes of this paper ^are to

investigate

the Dirichlet

problem

^{for the}

elliptic equation

in a

half-space

^{and a}

complement

^{of a} bounded open set. We shall refer to the second

problem

^as the exterior Dirichlet

problem.

Given an open set Q c we denote

by

^the Banach space of functions u in

Z2(S~) having

^weak

(distributional)

derivatives

Diu.

(i = 1, ..., n)

ⁱⁿ

L2(Q).

^{A norm is} introduced

by defining

The closure of in

Wl,2(Q)

is denoted

by

^A local space consists of functions

belonging

^to for every bounded open ^set ,~’ such that S~.

To motivate our

approach

^to the Dirichlet

problem

^{assume for}

simplicity

^{that .~ is}

uniformly elliptic

^{and the} coefficients ^c

(*)

Indirizzo dell’A.:

Department

^of Mathematics,

University

^{of Queens-} land, St. Lucia Queensland 4067, ^Australia.

and

f

^are measurable and bounded on SZ. A function u is said to be

a weak solution of the

equation (1)

^{if u E and u satisfies}

for every ^v

EWl,2(Q)

^with

compact support

^{in ,~.} Let q ^E

_L2(aS2)

and assume that there is a function ffJl ^E

Wl,2(Q)

such that q ⁼ ffJ1

on a,~ in the sense of trace. A weak solution in of the equa- tion

(1)

^{is a} solution of the Dirichlet

problem

^{with the}

boundary

^condi-

tion u = 99 ^on 8Q if The basic results

concerning

the Dirichlet

problem

in W1,2-framework can be found in

Ladyzhens- kaja

and Ural’ceva

[16], Gilbarg

^and

Trudinger [10]

^and

Stampacchia [24], [25].

^In the above definition it is assumed that the

boundary data

^{is a trace of some function}

belonging

^to

Wl,2(Q).

This condi- tion is rather

restrictive,

^because not every function in

L 2

^{is the}

trace of some function

belonging

^to

(see

Lions and

Mage-

nes

[17]

Theorems 7.5 and 9.4

Chapter 1).

It is clear that the Dirichlet

problem

^with

L2-boundary

^data

requires

^{a new} definition. The first

attempt

^to define the Dirichlet

problem

^with

L2 -boundary

^{data has}

been made

by

Mikhailov who in a series of articles

[11], [18], [19]

and

[20]

examined this

problem

^{in a bounded} domain under the _assump- tion _aij E

CI(D) and bi,

^{c E}

(see

^{also Necas}

^[22]

^and

[23]).

^Similar

results were also obtained

by Kapanadze [15].

The author and

Thomp-

son

[5]

extended Mikhailov’s results to the

equation

with coefficients

satisfying

^some

general integrability

conditions. In the articles ^men- tioned above the

boundary

^8Q

belongs

^to C2. For the

Laplace equation

the Dirichlet

problem

^with

L2-boundary

^{data was solved in bounded}

Lipschitz

^domains

(see Dahlberg [8],

Jerrison and

Kenig [12], [13]).

We mention here that Jerrison and

Kenig

also extended their results to bounded non-smooth domains for an

equation

^with C’-eoefficients

(see [14]).

The

plan

of this paper is ^as follows. In sections 1-6 we examine the Dirichlet

problem

^{in a half} space. The main result is an energy estimate

(section 3,

^the

inequality (19))

^{for the}

equation

^~u

+

^{lu =}

f,

where I is a

sufficiently large parameter.

^Next

applying

^{the result}

of Bottaro and Marina

[3]

^{we establish the existence of a}

unique

^solu-

tion to the Dirichlet

problem

^with

L2-boundary

data for the

equation (1)

with the condition

e(x) ~

^Const &#x3E; 0. In sections

7,

^{8 and 9 similar}

results are established for the exterior Dirichlet

problem.

^Methods

used in both cases are similar and

presented

^{in some} details. We

point

^{out that the}

assumptions

^{on the} coefficients _ai~ for the Dirichlet

problem

ⁱⁿ the half space ^are

considerably

weaker than the

correspond- ing assumptions

^{for the} exterior Dirichlet

problem.

^{This follows from}

the fact that the

boundary

of the half space is flat.

Among

the papers devoted to

study

^the

boundary

^value

problems

in unbounded domains

we mention the work of Benci and Fortunato

[2],

ⁱⁿ which the Dirichlet

problem

^{with zero}

boundary

^{data in a}

weighted

^Sobolev space has been studied.

In this paper ^we make

frequent

^use of the Sobolev

inequality

where

The most

significant

feature of this

inequality

lies in the

independence

of the constant S of the domain S~ which makes

possible

^{to use it in}

unbounded domains

(see

^Federer

[9]).

2. Traces.

Let

Rt

⁼ &#x3E;

01.

^{We denote a}

point

^{x E}

Ri by x

⁼

= (x’,xn),

^{where x’= ... , I}

Throughout

^{sections 2-6 we make the}

following assumptions

^about

the

operator

^{L :}

(A)

^{.L is}

uniformly elliptic

ⁱⁿ

.Rn , i.e.,

^{there exists a}

positive constant y

^{such that}

for all

and $ e Rn ,

^moreover

(i, j = 1, ..., n).

(B) (i)

There exist

positive

constants x and 0 a 1 such that

for all and all

oo) .

All constants in this paper will be denoted

by ^{C~ .}

The statement

«

C, depends

^on the structure of the

operator

1)&#x3E; means that

Ci depends

on n, a,

b, k,

m, and the norms of the coefficients

Daij,

ai;,

bi

^and

c in

appropriate

spaces. ^co

In the

sequel

^we shall need the

following elementary

^lemmas.

is bounded

independently of 6

^E

(0, T/2].

PROOF.

Integrating by parts

Denoting

^{the last}

integral by

^{J and}

applying Young’s inequality

we obtain

Taking s

⁼

(1

^-

y)/4

the result follows.

We also need the

following simple

observation.

LEMMA 3. let be a solution

of (1 ).

^Then

for

every

where ^a

positive

^{constant C}

depends

^{on the structure}

of

^the

operator

^{Land r.}

PROOF. Let v ⁼

U02

^{where 0 E}

Using

^{v as a} test function in

( 2 )

^{we obtain}

It follows from

ellipticity

of L and the

inequalities

^of

Young

^and

Sobolev that

where a

positive

constant C

depends

^on the structure of the

operator

JD. Here we have used the fact that c ⁼ c1

+

C2 with _Cl E J~ and _c, E

To

complete

^the

proof put 0 = 0, ,

^where

^0,

^{is an}

increasing

sequence

of

non-negative

functions in with the

gradient

bounded inde-

pendently

^{of v and}

converging

^{as v --~ oo to a}

non-negative

^function

f on

lit equal

^{to 1 for and}

vanishing

^for

THEOREM 1. be ^a solution

of (1)

ⁱⁿ

~.

^{Then the}

following

conditions are

equivalent :

PROOF. Let

0 C 38o C 1.

^{Define a}

non-negative

^function

q e C2 ([0, oo))

^{such that = xn for and}

21(x,,)

^{= 1 for}

We may ^assume that

q(rn) &#x3E; 6

^{for all and 0}

^{6 60.}

Let

where 0 is a

non-negative

function in Since for

every 6

C x~

v( ~, xn)

^{has a}

compact support

^{in it follows} from Lemma 5 that

v is an admissible test function in

(2),

^hence

Denote the

integrals

^{on the left hand side of}

(4) by J1,

...,

Je.

^{It follows}

from the

ellipticity

^{of .~ that}

By Young’s inequality

where a

positive

^constant

⁰¹ depends

on y ^and

Similarly

where ^a

positive

^constant

C, depends

^on

11 bi ll,..

^Now

^according

^to

the

assumption (B iii)

where _°1 E and ₀₂ E

By

^Hölder’s

inequality

where

1/2*

⁼

1 /2 - 11n.

^Now

by

^Sobolev’s

inequality

were 8

depends only

^{on n. Since we} may ^assume

we obtain

by Young’s inequality

where a

positive

^constant

03 depends

on n, and _y.

By

^{Green’s formula we have}

and

by

^the

assumption (B ii)

where a

positive

^constant

C,~ depends

^{on and xi .}

Integrating

by parts

By

^the

assumption (Bi)

^{we obtain}

where a

positive

^constant

C, depends

on x, y and From the last

inequality

^we deduce the

following

estimate for

J.,

where ^a

positive

^constant

⁰⁶ depends

^{on x, and}

. ^Inserting

the estimates

(5)-(10)

^into

(4)

^{we obtain}

If the condition

(I)

holds then

by

^{Lemma 1 the}

integrals

are bounded

independently

^{of 6. Now}

put 0 = 0,, ^{where 0,}

^{is an}

increasing

sequence of

non-negative

^{functions in}

tending

^to

1 as v ^{- oo} with the

gradient

^bounded

independently

^{of v.}

Letting

v ^{- oo} in

(11)

it follows from Lemma 3 that

The

implication

^{o I ~ II »} follows from Lemmas 1 and 3 and the

Lebesgue

convergence theorem.

To show that « II =&#x3E; I » observe that

Now

by

^{Lemma 2} the condition

(II) implies

^{that the}

integrals

are bounded

independently

^{of 6.}

Repeating

^the

argument

^{from the}

step

«I =&#x3E;Tl)) the result follows.

REMARK 1. It follows from the

proof

^{Theorem 1} that the condi- tion

(II) implies:

As an immediate consequence ^we obtain ^, COROLLARY 1..Let u E be a solution

o f (1)

ⁱⁿ

that one

of

the conditions

(I)

^or

(II)

holds. Then there exists ^a

function

q ^{E and a} sequence

6,

^- 0 as v ^{- oo} such that

for

every

THEOREM be a solution

o f (1)

ⁱⁿ

R:. Suppose

that one

of

th e conditions

( I )

^or

( II )

holds. Then there exists ^a

function

~ ^E such that

for every Y E L2(Rn-1).

PROOF. Since

~)2

^{dx’ is}

bounded,

say and

ann(x’, 6)

~n-l

continuous and bounded it follows from

Corollary 1,

^that

It suffices to show the existence of the

and

v(x)

^{= 0}

elsewhere, where 77 oo))

^is

^the

function introduced in the

proof

of Theorem 1.

Taking v

^{as a test function in}

(2)

^{we obtain}

Since for every 0

y

¹

"ri "n

and Y has a

compact support,

^the

Lebesgue

dominated convergence theorem

implies

^the

continuity

^of

Now

By

^the

Lebesgue

^dominated convergence theorem and the

previous

part

^{of the}

proof,

^the

right

hand side of the last

inequality

^{tends to}

0 as b ^- 0 and this

completes

^the

proof.

Our next

objective

^{is to establish the}

L2-convergence

^of

u( ~, ð)

to 99 as 6 ^- 0. To do this we first show that the norm of

u( ., ð)

converges to the norm of 99. The result then follows

by

the uniform

convexity

of _{the space}

L2.j

THEOREM 3. Let U E

IVI,2 (R,+,)

^{be a solution}

of (1 ). Suppose

^that

one

of

^the conditions

(I)

^or

(II)

holds. Then there exists a

function

99 ^E L2

(Rn_1)

^{such that}

PROOF. If Il E

1’1’ ~’Z~lLn~~

^then

by

^the

argument

^{used in the}

proof

of Theorem 1 we obtain

where q

is the function introduced in the

proof

of Theorem 1. Define

it is clear that ^. Thus

We shall show that

.and

(15)

Indeed

Here we have used the fact that u is ^a solution of

(1)

^{and the}

identity (2)

with the test function

where 27

is the function introduced in the

proof ’of

^{Theorem 1.}

To prove

(15)

observe that

.

Using

^the conditions

(I), (II)

and Lemma 1 one can show that We

only

restrict ourselves to the term

J,.

Integrating by parts

^{the term}

J3

^{can be written} in the

following

^form

Hence

by

^the

assumption (B ii)

where a

positive

constant C

depends

^on xl, n and

(~== 11

...I

n -1). By

^{Lemma 1 the first}

integral

^{on the}

_right

hand side tends to 0 as 6 -~ 0. Now

by

^H61der’s

inequality

It

follows

from the condition

(I)

^that

and

consequently by

the condition

(II)

the second

integral

^{on the}

right

^{hand side of}

(16)

^{also tends to 0 as 6 - 0. It follows from}

(~.3),

(14)

^and

(15)

^that

To

complete

^the

proof

observe that the norms

are

equivalent

ⁱⁿ

3.

Energy

^estimate.

Consider the

elliptic equation

of the form

in where A is a real

parameter.

The results of section 2

suggest

^the

following

definitian of the Dirichlet

problem.

A weak solution

is a solution of the Dirichlet

problem

^{with the}

boundary

^condition

The main result of this section is the

following

energy estimate.

THEOREM 4. Let ^I be a solution

of

^{the Dirichlet}

problems

(17), (18).

Then there exist

positive

^constants

d, Ao

^{and C}

independent

of

’11, such that

for all

A&#x3E;

^{where a}

positive

^{constant C}

depends

^on the structure of L.

PROOF. Since is a solution of the Dirichlet

problem

for certain

30

&#x3E; 0 and

consequently by

^{Theorem 1 the condition}

(II)

^holds. All constants

appearing

ⁱⁿ

the

proof

^are

indepenent

^{of 8 and}

depend

^on the structure of L and may vary from line to line.

By

^{the same} considerations as in the

proof

of the

step

^I =&#x3E; II » of Theorem 1 we show that

~o, whereq

^{is the} function introduced in the

proof

of Theorem 1.

Letting 6

^{- 0 we deduce} from the last

inequality

It

follows

from

(20)

^and

(3) (see

^Lemma

3)

^that

Now

adding

^the

inequalities (3)

^and

(21)

^we

get

By

^{a similar manner}

(see

^the

proof

^{of the}

step

^{« I} =&#x3E; II » of Theorem

1)

we obtain

Now

inserting (22)

^into

(23)

^{we obtain}

Now we make use of the

following

fact: for every

0 ~ ~O

^{1 and 0}

^d

¹

Thus

(19)

follows from

(22)

^and

(23) provided A

^is

sufficiently large

and d

sufficiently

^small.

. Dirichlet

problem

ⁱⁿ

Ri.

We are now in a

position

^{to establish the existence of a solution}

to the Dirichlet

problem.

THEOREM

~, ~ ~,o

^{Assume that r1}

(i =1,

..., n)

^{and that Then}

for (Rn_i )

^there

exists a

unique

^solution

of

the Dirichlet

problem (17), (18)

ⁱⁿ

PROOF. Let

(qm)

^{be a} sequence of functions in

converging

in

L2(Rn_i)

^{to the} function 99. Put

~rn ⁼

1, 2, ....

^{. It follows from}

[3]

that there exists a

unique

^solution

in

&#x3E;

^{of the Dirichlet}

problem

By

^{Theorem 4}

for where C is a

positive

^constant

independent

of p and q. Con-

sequently

^{is the}

Cauchy

sequence in the ^norm

and the result follows.

THEOREM 6.

Suppose

^{that the}

essumptions of

Theorem 5 hold and

moreover 0 Then

for

^there

exists ^a

unique

^{solution to} the Dirichlet

problem (1), (18)

ⁱⁿ

and moreover

where C is a

positive

^constant

depending

^{on the structure}

of

^the

operator

^L.

PROOF. Let

lo

^{be a}

sufficiently large positive

^constant.

By

^Theorem

5 the Dirichlet

problem

has a

unique

solution in

loc n satisfying

the energy estimate

(25).

On the other hand in virtue of Theorem 1.1 in

[3]

the Dirichlet

problem

has a

unique

^{solution in}

~W1~2(.Rn )

^{and moreover}

where C is a

positive

constant. Thus the function u ⁼ _~o

-f- v

^{is a} solution of the

problem (1), (18)

^{and it is} obvious that

where C is a

positive

^{constant. To} obtain the estimate

(25)

^{it suffices}

to derive the estimate of the form

(23)

for sup

~)2

dx’ and then

Rn_1

use the final

part

^{of the}

argument

^{of the}

proof

of Theorem 4 and the

inequality (27).

The next result establishes the relation between the solution of the Dirichlet

problem

^{in and}

T~1’2(.Rn).

^We

point

^{out here}

that

by

^{a solution} of the Dirichlet

problem

^{in we mean a}

solution of

(1)

^{with the}

boundary

^{condition in the sense of trace}

(see Introduction).

THEOREM 7.

Suppose

^{that the}

assumptions of

^{Theorem 6}

hold,

^that

and that there exists a

f unction

^{such that}

q;l(X’, ^{0 )}

^_

~p(x’ )

^on

^{Rn-l in}

^{the sense}

o f

^trace. Then the solution

o f

^the

Dirichlet

problem (1), (18)

^{in solution}

of

^{the same}

problem

in

PROOF. It follows from

[3]

^that the Dirichlet

problem

^{Zu =}

f

on

.Rn

^{and u = y on}

^Rn-l

^{has a} solution in which is also a

solution in The result follows from the

uniqueness

^{of the}

solutions in of the

problem (1), (18).

5.

Weighted

estimate of the

gradient.

It is well known that a solution of the Dirichlet

problem

^{for the}

Laplace equation

where q;

EL2(Rn-t)

satisfies the

following

^relation

-"n

The

question

^arises whether the

inequality,

RTn

remains true for the solution of the

problem (1), (18)

ⁱⁿ

c (see ^[26]

^p.

^82-83).

^The

^following

theorem contains

a

partial

^{answer to this}

question.

THEOREM 8.

Suppose

^that

c(x) &#x3E; Const

&#x3E; 0 orc ₇ c E

+ + bi

^{= 0}

(i

⁼

1,

...,

n)

^and

f

^{= 0 on}

X -

^{Let u be a solution}

of

the Dirichlet

problem (1), (18)

^{in Then}

where a

positive

^{constant C}

depends

^{on the structure}

of

^the

operator

^L.

PROOF. Let be an

increasing

sequence of functions in

C2([o, oo))

converging

^to rn ^on

[0, c&#x3E;o)

^with

properties: i7,(x,,)

^{= xn for}

q(rn) &#x3E; 3

^for xn &#x3E;

2 60

^and

~D2~v ~

^{are bounded}

indepen- dently

^{of v.} q, may be chosen ^to be constant for

large

Xnè ^i,

Taking

as a test function in

(2)

^{we obtain}

We denote the sum of the second and third

integral by J1

^and

integrat-

ing by parts

^{we obtain}

where a

positive

^constant

C, depends

^on m.,, sup

ID*17,1

^and

sup

ID2,7,1. Similarly

^{the fourth}

integral Jg

^{can be estimated} in the

following

way

where a

positive

^constant

O2 depends

^on "1’ and

supID77,1.

₊

Finally using

^the

ellipticity

^{and the fact that c} is bounded below

by

.a

positive

constant we obtain

etting

^{v - oo and 6 - 0 we derive} from the last

inequality

where

positive

^constants

C3

^and

Õ3 depend

^{on the structure} of the opera- tor .~. Now

applying

the energy estimate

(25)

^{we obtain}

(28).

REMARK 2. The estimate

(28)

^can be extended to the nonhomo-

I

geneous

equation provided

1

6. Estimate of derivatives of’ the second order.

In this section we

replace

^the

assumptions (B i)

^and

(B ii) by

^the

following

^condition

( k, i, j,

⁼

1,...,’n),

^where

b and x1 are positive

constants.

THEoRF,3i 9. u be a solution in

loc of

^the

problem (1), (18).

T hen

where C is ^a

positive

^constant

depending

^{on the structure}

o f

^L.

PROOF. It follows from Theorem 8.8 in

[10] (p.173) that u

^E

We shall first show that for every ^r &#x3E; 0

Indeed, taking v

⁼

Dkw,

^{where w E with}

compact support,

as a test function in

(2)

^and

integrating by parts

^{we obtain}

Now let w ⁼ where 0 is ^a

non-negative

function in such that W ⁼ 0 for and

xERn-l.

Then

Applying

^the

inequalities

^of

H61der, Young

and Sobolev we deduce from the last

equation

^that

where ^a

positive

constant C

depends

^on the structure of L. Now

put

where

(Py

is an

increasing

sequence of

non-negative

^functions

in with

supports

^{in on}

and with bounded

independently

^{of v.}

Letting

^{v - 00 the ine-}

quality (30)

follows. To establish

(29)

^let

where 77

^{is a function} introduced in the

proof

of Theorem 1.

By (30)

~,u is an admissible test function in

(31)

^and

We may ^assume

Applying

^the

inequalities

^of

Young,

Holder and Sobolev ^we

easily

derive from the last

inequality

the follow-

ing

^estimate

where C is a

positive

^constant

depending

^on the structure of the opera- tor

L,

and the result follows.

7.

Exterior Dirichlet problem.

Preliminaries.

Let Q c

Rn

^{be a}

complement

^{of a bounded closed set with}

boundary

8Q of class C2. we consider the

equation (1).

^{Let x E}

Q,

^{we denote}

by r(x)

^{the distance from x to 3D.}

We make the

following assumptions

There exists a

positive

constant y such that

for all x E .~

and ~

^moreover aij ^E

L’(92)

^r1

(i, j

⁼

1,

...,

n).

We also assume that

xr(x)-a (i, j

⁼

1,

... ,

n)

^{in some}

neigh-

bourhood N of the

boundary 8Q,

^{where x} &#x3E; 0 and 0 a 1 are cons- tants and that

Dkaii N) (k, i, j

⁼

1,

...,

n).

(~2) bi

^E

f

^E

22(Q)

^{and c E}

+ Ln(,Q) (i

⁼

1,

...,

n).

It follows from the

regularity

^{of the}

boundary

^8Q that there is

a number

~o &#x3E; 0

such that the domain

with

boundary

possesses the

following property:

to each

ro e 8Q

^{there is a}

unique point ro(so)

⁼ xo ^- where

v(xo)

^is the outward normal to 8Q _{at xo .}

According

^{to Lemma 1 in}

[10]

p.

382,

^{the distance}

r(x) belongs

^to

C2(Q -

^if

6,

^is

sufficiently

small. We extend

r(x)

^as

positive

function of class

C2(Q)

^and

denote this extension

again by r(x).

^We may ^assume that N c Q -

Do..

Let xa denote an

arbitrary point

^of

^8Qd.

For fixed 6 E

(0,

^let

and

where

IA I

denote the

(n - 1)-dimensional

^{Hausdorff measure of}

A,

and

vo(xo)

^{is the outward normal to at xo. Mikhailov}

[19] proved

that there is a

positive

number Yo such that

We will use the surface

integral

where u E and the values of

son

^are understood in the sense of trace.

Let

Ro

^{be a}

positive

^{number such that}

Qo

^c

.R~

^for

every R &#x3E;

Ro .

^Set

°

for all

..R &#x3E; .Ro .

In the

sequel

^we shall need the

following

estimate: if

and p

^{is a} constant in

[0,1),

^then

for 6 E

(0, ðo/4],

where .~1 is a

positive

^constant

independent

^{of 6 and u.}

This

inequality

^{can be} established

by

^{the same}

argument

^{as in the}

proof

of Lemma

2;

^{in the}

proof

^{we use the}

inequality (32).

Moreover it is _easy to see that if

M(6)

^{is bounded on}

(0, ~o]

^then

for every p ^E

[0,1)

for all 6 E

(0, bo/21,

^where

^M1

^{is a}

positive

^constant

independent

^of

~ and u.

THEOREM 10. Let u be ^a solution

of (1 ) belonging

^to

Wlo~(SZ).

^Then

the

following

conditions are

equivalent

PROOF. We

only

^{sketch the}

proof

^{because it} is identical to that of Theorem 1 in

[5].

^{Fix an} .R &#x3E; and let 0 be a

non-negative

^func-

tion in such that ⁼ 1 for

lxl ^c.R

^and

Ø(x)

^{= 0 for}

Ixl

&#x3E; 2R.

Suppose

^that holds. Thus u E

)

^for

every

&#x3E;

.Ro

^and

(34)

^{holds. It} is clear that

is an admissible test function in

(2)

^and

By

^{Green’s formula}

(see ^[21],

^p.

139)

^{we have}

and

By

^the

ellipticity

^{condition and the}

inequalities

^of

Young

and Sobolev

we derive from

(35), (36)

^and

(37)

^that

Now choose q ^_

0,,

^where

(Wr)

^{is an}

increasing

sequence of ^non-

negative

^{functions in}

converging

^{to 1 for with}

IDØ,I

bounded

independently

^{of v.}

Letting 6 - o

^{and v - oo} the result

easily

^follows.

Now suppose that the condition

(Ill)

^holds.

By (35)

^and

(36)

we have

Using (33)

^{and the}

inequalities

^of

Young,

Holder and

Sobolev,

^it

is _{easy to} see that

M(6)

^{is bounded on}

(0, 5o].

Our next

objective

^{is to} prove that u has ^{a trace on} aS~ in

L2(ôQ).

We first state the

following preliminary result,

^{which is} easy ^to prove

(see

^{Theorems 2 and 3} of Section

2).

THEOREM 11..Let be a solution

of (1). Assume

^that

one

of

the conditions

(Ii)

^or

(III)

holds. Then there exists a

function

such that

for

every g ^E

_L2(aQ).

To prove that ^- 99 ⁱⁿ

L2(ôQ)

^{we show that}

and the result follows from the uniform

convexity

^of

~SE

_ _

Fix .R &#x3E;

.Ro,

^E

(0, ~o]

^we define the

mapping ^lJ2R

^-~

by

where

xa(x)

^denotes the nearest

point

^{to x on and =}

xa~2(x)

for each x E 3D. Moreover &#x3E;

5/2

^{and x8 is}

uniformly Lipschitz

continuous. Note that if u E then

u(r’)

^{E for each}

..R &#x3E;

Ro .

To prove

L2-convergence

^{of we shall need the}

following

^tech-

nical lemmas

LEMMA 4.

I f rif

^{and then}

5.

If gEL2(QR)

^and

r!fEL2(QR)

and suppose that

is bounded ^on

(0, ~o],

^{then a.,}

with inner

product (norm)

^denoted

with inner

product (norm)

^denoted

THEOREM 12. Let ^u E

loc

^{be a solution}

of (1 )

such that one

of

the conditions

(11)

^or holds. Then there is a

function

99 c such that

u(xa)

converges ^to 99 ^E

PROOF. The

proof

^{is similar to} that of Theorem 4 and is the

repeti-

tion of the

argument given

^{in the case} of bounded domain

(Theorem

⁴

in

[~]) .

Since

B1.BB1

^and

11.112

^are

equivalent

it sufficient to show that there

is 99

^E

L:

² such that lim

u(xi)

⁼ 99 ^{in 2}

By

^uniform

convexity

^of

.~2

^{it suffices to show that}

By

Green’s theorem ^we find that

Let 0 be defined as in the

proof

of Theorem

10,

^then

Consequently

^{we obtain}

for 6 E

(0, 30],

^{since = x on}

,SZa,2R

^{and = 1 on}

^DR.

^{We show}

that

and

since ^{= on}

aS2.

The relation

(38)

^follows from Lemmas 4 and 5. To establish

(39)

we use

(2)

^{with a test} function v

given by

^the

following

^formula

Theorem

12 justifies

^the

following

definition of the Dirichlet

problem.

A weak solution of

(1)

^{is a solution}

of the Dirichlet

problem

^{with the}

boundary

^condition

B.

Energy

^estimate for the exterior Dirichlet

problem.

Assume that the distance function

r(x)

is extended into S~ in such

a way that

on &#x3E;

Ro),

^where

1~1

^{is a}

positive

^constant.

Let q~

^{E and consider the}

following

^Dirichlet

problem

ⁱⁿ

where A is a real

parameter.

^If

A&#x3E; 10 (10 sufficiently large)

^then

^by

Theorem 6 in

[5]

there exists ^a

unique

^{solution. The}

boundary

^condi-

tions

(44)

is understood in the ^sense of

L2-convergence and by (43)

we mean that

Wlt2(QR)

^{for every} function P E such that P =1 in some

neighbourhood

^of

Ix

^{_ Rand P = 0 on Q -}

LEMMA 6.

Suppose

^{that r}

satisfies

^{the condition}

(E)

^{and let u be a}

solutions in

loc ) of

the Dirichlet

problem (42), (43)

^and

(44).

^Then

there exist

10

&#x3E; 0

(sufficiently large)

^{and d} &#x3E; 0

(sufficiently small)

^such

that

for

^all

R &#x3E; .Ro,

^where

positive

^constant

Âo,

^{d and C}

depend

^{on the struc-}

ture

of

^the

operator

^{and acre}

independent o f

^R.

PROOF. The energy estimate

(45)

^was

essentially proved

ⁱⁿ

[5].

We

repeat

^the

proof

^to show that the constants

0,

^{d and}

Ao

^are

indepen-

dent of R. Let

on

elsewhere .

By

^{Green’s theorem}

Now

applying Holder

^y

Young

^{and Sobolev}

inequalities

^{we obtain}

-where a

positive

^constant

Ci

^is

independent

of .Z~. On the other hand

we have

Letting 8 -

^{0 in}

(47)

^{we obtain}

It follows from

(48)

^and

(49)

^that

-where

O2

^and

03

^are

positive

^constants

independent

^of .R. Now observe that

by

^the

assumption (E)

^{we have}

where and

where and

consequently

Similarly

Choosing

^d

sufficiently

^{small and}

Ao sufficiently large

^the result follows from

(48), (49), (50), (51)

^and

(52).

9. Existence of a solution of the exterior Dirichlet

problem.

It is clear that one can deduce from Lemma 6 the existence of ^a solution of the Dirichlet

problem (42), (43)

^and

(44)

ⁱⁿ

-Wl,2

As an immediate consequence of Lemma ^{6 we} obtain

THEOREM 13.

Suppose

that the distance

f unction

^r

satisfies

^{the condi-}

tion

(E) . Let

99 ^E and

f (x)2r(x)

^{dx oo. Then}

^for

^every

^{À;;;. Ào}

there exists ac

unique

^solution

of

the Dirichlet

problem (42), (40)

^and

(41)

in and moreover

where ^a

positive

^{constant 0}

depends

^{on the structure}

of

^the

operator

^L.

PROOF. For fixed .R &#x3E;

.Ro

^{consider a}

solution vR

^{of the}

problem (42), (43)

^and

(44)

^{and set}

vR(x)

^{= 0 for}

lxl

&#x3E; .R. It follows from the energy estimate

(45)

that there exists a sequence vR

tending weakly

^to

a solution of the

problem (42), (40)

^and

(41).

From now on introduce the

following assumption

^{on the distance}

function r

(F)

The distance function

r(x)

^on is extended to a function in

C2(,~)

^{such that}

r(x)

^{= 1 on S~ n}

~ .Ro~.

It is obvious that condition

(F) implies (E).

THEOREM 14. In addition to the

hypotheses

^and

(A2)

^assume

that

bi

^E

(i

⁼

1,

...,

n),

^{c E}

+

^{and c} &#x3E; Const. &#x3E; 0

on Q. let and Then there exists a

unique

^solu-

tion u

of

the Dirichlet

problem (1 ), (40)

^and

(41)

^{in and moreover}

where a

positive

^{constant C}

depends

^{on the structure}

o f

^Z.

The

proof

is similar to that of Theorem 6 and therefore is omitted.

We

only point

^{out that we}

again

^use here the results of Bottaro and Marina

[3].

Under our

assumption

^a solution in of the

problem (1), (40)-(41) belongs

^to

By

^{the same}

argument

^{as in the}

proof

of Theorem 9 one can establish the

following

^{estimate of the derivative}

of the second order of a solution in of the

problem (1), (40)

and

(41).

THEOREM 15.

Suppose

^{that the}

assumptions of

^{Theorem 14 hold.}

Let ’U be a solution in

of

^the

problem (1), (40)

^and

^(41).

^Then

where C is a

positive

^constant

depending

^{on the structure}

of

^the

operator

^~.

We mention here that the estimate of the derivative of the second order of a solution of the exterior Dirichlet

problem

ⁱⁿ

~2~2(S~)

^was

obtained

by

^Acanfora

[1].

10. Case

c ~ 0.

To establish the existence and the

uniqueness

^{of a solution of the}

Dirichlet

problem

^we have assumed that c ~ Const &#x3E; 0. If the coeffi- cient c is

only non-negative

^{one can} also construct a solution but

belonging

^{to a} different function space.

Namely

introduce the space

~1,2(Rn ) ^equipped

^{with the norm}

and

similarly

for the exterior Dirichlet

problem

the space

Jf’1,2(Q)

with the norm

where r s atisfies the condition

(.F’). By

^{Theorem 7 a solution of the}

Dirichlet

problem (1), (18) belongs

^and

by

^{Theorem 14 a}

solution of the exterior Dirichlet

problem (1), (40)

^and

(41) belongs

to

y~l~ 2(,~)

^.

Now denote

by (D(Q))

^the

completion

^of

with

respect

^{to the norm}

By

^Sobolev’s

inequality (D(S2) c ~2*(S~)),

^whence

c

(-D(S2)

^c

THEOREM 16.

Suppose

^{that the}

assumptions (A)

^and

(B )

^{hold. Let}

f

^e

.L2 (.I~ ) a’nd lfJ

^{e and moreover assume that}

b

^e

(i

⁼

1, ..., n),

^{and on} Then there exists a sol’U- t’ion ’U to the Dirichlet

problem (1), (18), belonging to the

space

W1,2(R+n) +

+

Here the

boundary

^condition

(18)

^{is satisfied in the}

following

sense:

for

every .R

&#x3E; 0

PROOF. Consider the Dirichlet

problem

where the

boundary

condition is understood inthe sense of L2-conver- gence

(see

^section

3).

^If

Ao

^is

sufficiently large

^then

by

^{Theorem 5}

there exists a

unique

^{solution uo}

belonging

^to the space

T~1’2(.Rn ).

On the other hand

by

the result of Chicco

[7]

the Dirichlet

problem

has ^a solution in

D(X).

^{It is} clear that v

+

uo is a solution of

(1), (18)

ⁱⁿ

T~’1’2(.Rn ) -~- D(.Rn ). The L2-convergence

^{in the sense}

(54)

^follows

from the fact that

In a similar manner one can establish

THEOREM 17.

Suppose

^{that the}

assumptions (A1)

^and

(A2)

^hold.

.Let 99

^{E and} moreover assume that

bi

^E

(i =1,

...,

n),

c E

Ln/2(,Q)

^and

c(x) ~

^{0 on 92.} Then there exists a solution

of

^{the Dirichlet}

problem (1), (40)

^and

(41)

ⁱⁿ

+ D(SZ).

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